Convolutional Neural Networks

Handling image data

Joaquin Vanschoren, Eindhoven University of Technology

Overview

• Image convolution
• Convolutional neural networks
• Data augmentation
• Model interpretation
• Using pre-trained networks (transfer learning)

Convolution

• Operation that transforms an image by sliding a smaller image (called a filter or kernel) over the image and multiplying the pixel values
  • Slide an $n$ x $n$ filter over $n$ x $n$ patches of the original image
  • Every pixel is replaced by the sum of the element-wise products of the values of the image patch around that pixel and the kernel

# kernel and image_patch are n x n matrices
pixel_out = np.sum(kernel * image_patch)

horizontal_edge_kernel = np.array([[ 1,  2,  1],
                                   [ 0,  0,  0],
                                   [-1, -2, -1]])
diagonal_edge_kernel = np.array([[1, 0, 0],
                                 [0, 1, 0],
                                 [0, 0, 1]])
edge_detect_kernel = np.array([[-1, -1, -1],
                               [-1,  8, -1],
                               [-1, -1, -1]])

Demonstration

mnist_data = oml.datasets.get_dataset(554) # Download MNIST data
# Get the predictors X and the labels y
X_mnist, y_mnist, c, a = mnist_data.get_data(dataset_format='array', target=mnist_data.default_target_attribute); 
image = X_mnist[1].reshape((28, 28))
image = (image - np.min(image))/np.ptp(image) # Normalize

titles = ('Image and kernel', 'Filtered image')
convolution_demo(image, horizontal_edge_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r');

plt.rcParams['figure.dpi'] = 100
fig, axs = plt.subplots(3, 3)
titles = ('Image and kernel', 'Hor. edge filter', 'Filtered image')
convolution_full(axs[0,:], image, horizontal_edge_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
titles = ('Image and kernel', 'Edge detect filter', 'Filtered image')
convolution_full(axs[1,:], image, edge_detect_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
titles = ('Image and kernel', 'Diag. edge filter', 'Filtered image')
convolution_full(axs[2,:], image, diagonal_edge_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
plt.tight_layout()

Demonstration on Google streetview data

House numbers photographed from Google streetview imagery, cropped and centered around digits, but with neighboring numbers or other edge artifacts.

SVHN = oml.datasets.get_dataset(41081)
X, y, cats, attrs = SVHN.get_data(dataset_format='array',
    target=SVHN.default_target_attribute)

def plot_images(X, y, grayscale=False):
    fig, axes = plt.subplots(1, len(X),  figsize=(10, 5))
    for n in range(len(X)):
        if grayscale:
            axes[n].imshow(X[n].reshape(32, 32)/255, cmap='gray')
        else:
            axes[n].imshow(X[n].reshape(32, 32, 3)/255)
        axes[n].set_xlabel((y[n]+1)) # Label is index+1
        axes[n].set_xticks(()), axes[n].set_yticks(())
    plt.show();

images = range(5)
X_sub_color = [X[i] for i in images]
y_sub = [y[i] for i in images]
plt.rcParams['figure.dpi'] = 60
plot_images(X_sub_color, y_sub)

For recognizing digits, color is not important, so we grayscale the images

def rgb2gray(X, dim=32):
    return np.expand_dims(np.dot(X.reshape(len(X), dim*dim, 3), [0.2990, 0.5870, 0.1140]), axis=2)
Xsm = rgb2gray(X[:100])

X_sub = [Xsm[i] for i in images]
plot_images(X_sub, y_sub, grayscale=True)

Demonstration

plt.rcParams['figure.dpi'] = 100
def normalize_image(X):
    image = X.reshape((32, 32))
    return (image - np.min(image))/np.ptp(image) # Normalize

image = normalize_image(X_sub[3])
demo2 = convolution_demo(image, horizontal_edge_kernel,
                 vmin=-4, vmax=4, cmap='gray_r');

fig, axs = plt.subplots(3, 3)
titles = ('Image and kernel', 'Hor. edge filter', 'Filtered image')
convolution_full(axs[0,:], image, horizontal_edge_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
titles = ('Image and kernel', 'Diag. edge filter', 'Filtered image')
convolution_full(axs[1,:], image, diagonal_edge_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
titles = ('Image and kernel', 'Edge detect filter', 'Filtered image')
convolution_full(axs[2,:], image, edge_detect_kernel, vmin=-4, vmax=4, titles=titles, cmap='gray_r')
plt.tight_layout()

Image convolution in practice

• How do we know which filters are best for a given image?
• Families of kernels (or filter banks) can be run on every image
  • Gabor, Sobel, Haar Wavelets,...
• Gabor filters: Wave patterns generated by changing:
  • Frequency: narrow or wide ondulations
  • Theta: angle (direction) of the wave
  • Sigma: resolution (size of the filter)

Demonstration

from scipy import ndimage as ndi
from skimage import data
from skimage.util import img_as_float
from skimage.filters import gabor_kernel

# Gabor Filters.
@interact
def demoGabor(frequency=(0.01,1,0.05), theta=(0,3.14,0.1), sigma=(0,5,0.1)):
    plt.gray()
    plt.imshow(np.real(gabor_kernel(frequency=frequency, theta=theta, sigma_x=sigma, sigma_y=sigma)), interpolation='nearest')

plt.subplot(1, 3, 1)
demoGabor(frequency=0.16, theta=1.2, sigma=4.0)
plt.subplot(1, 3, 2)
demoGabor(frequency=0.31, theta=0, sigma=3.6)
plt.subplot(1, 3, 3)
demoGabor(frequency=0.36, theta=1.6, sigma=1.3)
plt.tight_layout()

Demonstration on the streetview data

# Calculate the magnitude of the Gabor filter response given a kernel and an imput image
def magnitude(image, kernel):
    image = (image - image.mean()) / image.std() # Normalize images
    return np.sqrt(ndi.convolve(image, np.real(kernel), mode='wrap')**2 +
                   ndi.convolve(image, np.imag(kernel), mode='wrap')**2)

@interact
def demoGabor2(frequency=(0.01,1,0.05), theta=(0,3.14,0.1), sigma=(0,5,0.1)):
    plt.subplot(131)
    plt.title('Original')
    plt.imshow(image)
    plt.subplot(132)
    plt.title('Gabor kernel')
    plt.imshow(np.real(gabor_kernel(frequency=frequency, theta=theta, sigma_x=sigma, sigma_y=sigma)), interpolation='nearest')
    plt.subplot(133)
    plt.title('Response magnitude')
    plt.imshow(np.real(magnitude(image, gabor_kernel(frequency=frequency, theta=theta, sigma_x=sigma, sigma_y=sigma))), interpolation='nearest')
    plt.tight_layout()

demoGabor2(frequency=0.16, theta=1.4, sigma=1.2)

Filter banks

• Different filters detect different edges, shapes,...
• Not all seem useful

# More images
image3 = normalize_image(Xsm[3])
image5 = normalize_image(Xsm[5])
image13 = normalize_image(Xsm[13])

image_names = ('3', '5', '8') # labels
images = (image3, image5, image13)

def plot_filter_bank(images):
    # Create a set of kernels, apply them to each image, store the results
    results = []
    kernel_params = []
    for theta in (0, 1):
        theta = theta / 4. * np.pi
        for frequency in (0.1, 0.2):
            for sigma in (1, 3):
                kernel = gabor_kernel(frequency, theta=theta,sigma_x=sigma,sigma_y=sigma)
                params = 'theta=%.2f,\nfrequency=%.2f\nsigma=%.2f' % (theta, frequency, sigma)
                kernel_params.append(params)
                results.append((kernel, [magnitude(img, kernel) for img in images]))

    # Plotting
    fig, axes = plt.subplots(nrows=9, ncols=4, figsize=(6, 12))
    plt.gray()
    #fig.suptitle('Image responses for Gabor filter kernels', fontsize=12)
    axes[0][0].axis('off')

    # Plot original images
    for label, img, ax in zip(image_names, images, axes[0][1:]):
        ax.imshow(img)
        ax.set_title(label, fontsize=9)
        ax.axis('off')

    for label, (kernel, magnitudes), ax_row in zip(kernel_params, results, axes[1:]):
        # Plot Gabor kernel
        ax = ax_row[0]
        ax.imshow(np.real(kernel), interpolation='nearest') # Plot kernel
        ax.set_ylabel(label, fontsize=7)
        ax.set_xticks([]) # Remove axis ticks 
        ax.set_yticks([])

        # Plot Gabor responses with the contrast normalized for each filter
        vmin = np.min(magnitudes)
        vmax = np.max(magnitudes)
        for patch, ax in zip(magnitudes, ax_row[1:]):
            ax.imshow(patch, vmin=vmin, vmax=vmax) # Plot convolutions
            ax.axis('off')
    
    plt.rcParams['figure.dpi'] = 80
    plt.show()

plot_filter_bank(images)
plt.rcParams['figure.dpi'] = 100

Another example: Fashion MNIST

plt.rcParams['figure.dpi'] = 200
# build a list of figures for plotting
def buildFigureList(fig, subfiglist, titles, length):
    for i in range(0,length):
        pixels = np.array(subfiglist[i], dtype='float')
        pixels = pixels.reshape((28, 28))
        a=fig.add_subplot(1,length,i+1)
        imgplot =plt.imshow(pixels, cmap='gray_r')
        a.set_title(titles[i], fontsize=6)
        a.axes.get_xaxis().set_visible(False)
        a.axes.get_yaxis().set_visible(False)
    return

subfiglist = []
titles=[]

for i in range(0,10):
    subfiglist.append(X_fm[i])
    titles.append(i)

buildFigureList(plt.figure(1),subfiglist, titles, 10)
plt.show()

Demonstration

plt.rcParams['figure.dpi'] = 100
boot = X_fm[0].reshape((28, 28))
image=boot
@interact
def demoGabor2(frequency=(0.01,1,0.05), theta=(0,3.14,0.1), sigma=(0,5,0.1)):
    plt.subplot(131)
    plt.title('Original')
    plt.imshow(image)
    plt.subplot(132)
    plt.title('Gabor kernel')
    plt.imshow(np.real(gabor_kernel(frequency=frequency, theta=theta, sigma_x=sigma, sigma_y=sigma)), interpolation='nearest')
    plt.subplot(133)
    plt.title('Response magnitude')
    plt.imshow(np.real(magnitude(image, gabor_kernel(frequency=frequency, theta=theta, sigma_x=sigma, sigma_y=sigma))), interpolation='nearest')

demoGabor2(frequency=0.81, theta=2.7, sigma=0.9)

Fashion MNIST with multiple filters (filter bank)

# Fetch some Fashion-MNIST images
boot = X_fm[0].reshape(28, 28)
shirt = X_fm[1].reshape(28, 28)
dress = X_fm[2].reshape(28, 28)
image_names = ('boot', 'shirt', 'dress')
images = (boot, shirt, dress)

plot_filter_bank(images)
plt.rcParams['figure.dpi'] = 100

Convolutional neural nets

• Finding relationships between individual pixels and the correct class is hard
• We want to discover 'local' patterns (edges, lines, endpoints)
• Representing such local patterns as features makes it easier to learn from them
• We could use convolutions, but how to choose the filters?

Convolutional Neural Networks (ConvNets)

• Instead on manually designing the filters, we can also learn them based on data
  • Choose filter sizes (manually), initialize with small random weights
• Forward pass: Convolutional layer slides the filter over the input, generates the output
• Backward pass: Update the filter weights according to the loss gradient
• Illustration for 1 filter:

Convolutional layers: Feature maps

• One filter is not sufficient to detect all relevant patterns in an image
• A convolutional layer applies and learns $d$ filter in parallel
• Slide $d$ filters across the input image (in parallel) -> a (1x1xd) output per patch
• Reassemble into a feature map with $d$ 'channels', a (width x height x d) tensor.

Border effects (zero padding)

• Consider a 5x5 image and a 3x3 filter: there are only 9 possible locations, hence the output is a 3x3 feature map
• If we want to maintain the image size, we use zero-padding, adding 0's all around the input tensor.

Undersampling (striding)

• Sometimes, we want to downsample a high-resolution image
  • Faster processing, less noisy (hence less overfitting)
• One approach is to skip values during the convolution
  • Distance between 2 windows: stride length
• Example with stride length 2 (without padding):

Max-pooling

• Another approach to shrink the input tensors is max-pooling:
  • Run a filter with a fixed stride length over the image
    • Usually 2x2 filters and stride lenght 2
  • The filter simply returns the max (or avg) of all values
• Agressively reduces the number of weights (less overfitting)
• Information from every input node spreads more quickly to output nodes
  • In pure convnets, one input value spreads to 3x3 nodes of the first layer, 5x5 nodes of the second, etc.
  • Without maxpooling, you need much deeper networks, harder to train
• Increases translation invariance: patterns can affect the predictions no matter where they occur in the image

Convolutional nets in practice

• ConvNets usually use multiple convolutional layers to learn patterns at different levels of abstraction
  • Find local patterns first (e.g. edges), then patterns across those patterns
• Use MaxPooling layers to reduce resolution, increase translation invariance
• Use sufficient filters in the first layer (otherwise information gets lost)
• In deeper layers, use increasingly more filters
  • Preserve information about the input as resolution descreases
  • Avoid decreasing the number of activations (resolution x nr of filters)

Example with Keras:

• Conv2D for 2D convolutional layers
  • 32 filters (default), randomly initialized (from uniform distribution)
  • Deeper layers use 64 filters
  • Filter size is 3x3
  • ReLU activation to simplify training of deeper networks
• MaxPooling2D for max-pooling
  • 2x2 pooling reduces the number of inputs by a factor 4

model = models.Sequential()
model.add(layers.Conv2D(32, (3, 3), activation='relu', 
                        input_shape=(28, 28, 1)))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))

Observe how the input image on 28x28x1 is transformed to a 3x3x64 feature map

• Convolutional layer:
  • No zero-padding: every output 2 pixels less in every dimension
  • 320 weights: (3x3 filter weights + 1 bias) * 32 filters
• After every MaxPooling, resolution halved in every dimension

model.summary()

Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
conv2d (Conv2D)              (None, 26, 26, 32)        320       
_________________________________________________________________
max_pooling2d (MaxPooling2D) (None, 13, 13, 32)        0         
_________________________________________________________________
conv2d_1 (Conv2D)            (None, 11, 11, 64)        18496     
_________________________________________________________________
max_pooling2d_1 (MaxPooling2 (None, 5, 5, 64)          0         
_________________________________________________________________
conv2d_2 (Conv2D)            (None, 3, 3, 64)          36928     
=================================================================
Total params: 55,744
Trainable params: 55,744
Non-trainable params: 0
_________________________________________________________________

Completing the network

• To classify the images, we still need a Dense and Softmax layer.
• We need to flatten the 3x3x36 feature map to a vector of size 576

model.add(layers.Flatten())
model.add(layers.Dense(64, activation='relu'))
model.add(layers.Dense(10, activation='softmax'))

Complete network

model.summary()

Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
conv2d (Conv2D)              (None, 26, 26, 32)        320       
_________________________________________________________________
max_pooling2d (MaxPooling2D) (None, 13, 13, 32)        0         
_________________________________________________________________
conv2d_1 (Conv2D)            (None, 11, 11, 64)        18496     
_________________________________________________________________
max_pooling2d_1 (MaxPooling2 (None, 5, 5, 64)          0         
_________________________________________________________________
conv2d_2 (Conv2D)            (None, 3, 3, 64)          36928     
_________________________________________________________________
flatten (Flatten)            (None, 576)               0         
_________________________________________________________________
dense (Dense)                (None, 64)                36928     
_________________________________________________________________
dense_1 (Dense)              (None, 10)                650       
=================================================================
Total params: 93,322
Trainable params: 93,322
Non-trainable params: 0
_________________________________________________________________

Run the model on MNIST dataset

• Train and test as usual (takes about 5 minutes): 99% accuracy
  • Compared to 97,8% accuracy with the dense architecture

from tensorflow.keras.models import load_model

model = load_model(os.path.join(model_dir, 'cats_and_dogs_small_0.h5'))
test_loss, test_acc = model.evaluate(test_images, test_labels)
print("Accuracy: ", test_acc)

history = pickle.load(open("../data/models/cats_and_dogs_small_0_history.p", "rb"))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

10000/10000 [==============================] - 3s 266us/sample - loss: 0.0355 - accuracy: 0.9902
Accuracy:  0.9902

Tip:

• Training ConvNets can take a lot of time
• Save the trained model (and history) to disk so that you can reload it later

model.save(os.path.join(model_dir, 'cats_and_dogs_small.h5'))
with open(os.path.join(model_dir, 'cats_and_dogs_small_history.p'), 'wb') as file_pi:
    pickle.dump(history.history, file_pi)

Cats vs Dogs

• A more realistic dataset: Cats vs Dogs
  • Colored JPEG images, different sizes
  • Not nicely centered, translation invariance is important
• Preprocessing
  • Create balanced subsample of 4000 colored images
    • 2000 for training, 1000 validation, 1000 testing
  • Decode JPEG images to floating-point tensors
  • Rescale pixel values to [0,1]
  • Resize images to 150x150 pixels

Data generators

• ImageDataGenerator: allows to encode, resize, and rescale JPEG images
• Returns a Python generator we can endlessly query for batches of images
• Separately for training, validation, and test set

train_generator = ImageDataGenerator(rescale=1./255).flow_from_directory(
        train_dir, # Directory with images
        target_size=(150, 150), # Resize images 
        batch_size=20, # Return 20 images at a time
        class_mode='binary') # Binary labels

for data_batch, labels_batch in train_generator:
    plt.figure(figsize=(10,5))
    for i in range(7):
        plt.subplot(171+i)
        plt.xticks([])
        plt.yticks([])
        imgplot = plt.imshow(data_batch[i])
        plt.title('cat' if labels_batch[i] == 0 else 'dog')
        plt.tight_layout()
    break

Since the images are larger and more complex, we add another convolutional layer and increase the number of filters to 128.

model = models.Sequential()
model.add(layers.Conv2D(32, (3, 3), activation='relu',
                        input_shape=(150, 150, 3)))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(128, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(128, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Flatten())
model.add(layers.Dense(512, activation='relu'))
model.add(layers.Dense(1, activation='sigmoid'))

model.summary()

Model: "sequential_1"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
conv2d_3 (Conv2D)            (None, 148, 148, 32)      896       
_________________________________________________________________
max_pooling2d_2 (MaxPooling2 (None, 74, 74, 32)        0         
_________________________________________________________________
conv2d_4 (Conv2D)            (None, 72, 72, 64)        18496     
_________________________________________________________________
max_pooling2d_3 (MaxPooling2 (None, 36, 36, 64)        0         
_________________________________________________________________
conv2d_5 (Conv2D)            (None, 34, 34, 128)       73856     
_________________________________________________________________
max_pooling2d_4 (MaxPooling2 (None, 17, 17, 128)       0         
_________________________________________________________________
conv2d_6 (Conv2D)            (None, 15, 15, 128)       147584    
_________________________________________________________________
max_pooling2d_5 (MaxPooling2 (None, 7, 7, 128)         0         
_________________________________________________________________
flatten_1 (Flatten)          (None, 6272)              0         
_________________________________________________________________
dense_2 (Dense)              (None, 512)               3211776   
_________________________________________________________________
dense_3 (Dense)              (None, 1)                 513       
=================================================================
Total params: 3,453,121
Trainable params: 3,453,121
Non-trainable params: 0
_________________________________________________________________

Training

• The fit function also supports generators
  • 100 steps per epoch (batch size: 20 images per step), for 30 epochs
  • Provide a separate generator for the validation data

model.compile(loss='binary_crossentropy',
              optimizer=optimizers.RMSprop(lr=1e-4),
              metrics=['acc'])
history = model.fit(
      train_generator, steps_per_epoch=100,
      epochs=30, verbose=0,
      validation_data=validation_generator,
      validation_steps=50)

Results

• The network seems to be overfitting. Validation accuracy is stuck at 75% while the training accuracy reaches 100%
• There are many things we can do:
  • Regularization (e.g. Dropout, L1/L2, Batch Normalization,...)
  • Generating more training data
  • Meta-learning: Use pretrained rather than randomly initialized filters

import pickle
history = pickle.load(open("../data/models/cats_and_dogs_small_1_history.p", "rb"))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Data augmentation

• Generate new images via image transformations
  • Images will be randomly transformed every epoch
• We can again use a data generator to do this

datagen = ImageDataGenerator(
      rotation_range=40,     # Rotate image up to 40 degrees
      width_shift_range=0.2, # Shift image left-right up to 20% of image width
      height_shift_range=0.2,# Shift image up-down up to 20% of image height
      shear_range=0.2,       # Shear (slant) the image up to 0.2 degrees
      zoom_range=0.2,        # Zoom in up to 20%
      horizontal_flip=True,  # Horizontally flip the image
      fill_mode='nearest')

Example

# This is module with image preprocessing utilities
from tensorflow.keras.preprocessing import image
plt.rcParams['figure.dpi'] = 120

train_cats_dir = os.path.join(base_dir, 'train', 'cats')
fnames = [os.path.join(train_cats_dir, fname) for fname in os.listdir(train_cats_dir)]

# We pick one image to "augment"
img_path = fnames[5]

# Read the image and resize it
img = image.load_img(img_path, target_size=(150, 150))

# Convert it to a Numpy array with shape (150, 150, 3)
x = image.img_to_array(img)

# Reshape it to (1, 150, 150, 3)
x = x.reshape((1,) + x.shape)

# The .flow() command below generates batches of randomly transformed images.
# It will loop indefinitely, so we need to `break` the loop at some point!
for a in range(2):
    i = 0
    for batch in datagen.flow(x, batch_size=1):
        plt.subplot(141+i) 
        plt.xticks([]) 
        plt.yticks([])
        imgplot = plt.imshow(image.array_to_img(batch[0]))
        i += 1
        if i % 4 == 0:
            break
        
    plt.tight_layout()
    plt.show()

We also add Dropout before the Dense layer

model = models.Sequential()
model.add(layers.Conv2D(32, (3, 3), activation='relu',
                        input_shape=(150, 150, 3)))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(128, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(128, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Flatten())
model.add(layers.Dropout(0.5))
model.add(layers.Dense(512, activation='relu'))
model.add(layers.Dense(1, activation='sigmoid'))

(Almost) no more overfitting!

history = pickle.load(open("../data/models/cats_and_dogs_small_2_history.p", "rb"))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Interpreting the model

• Let's see what the convnet is learning exactly by observing the intermediate feature maps
  • A layer's output is also called its activation
• We can choose a specific test image, and observe the outputs
• We can retrieve and visualize the activation for every filter for every layer

• Layer 0: has activations of resolution 148x148 for each of its 32 filters
• Layer 2: has activations of resolution 72x72 for each of its 64 filters
• Layer 4: has activations of resolution 34x34 for each of its 128 filters
• Layer 6: has activations of resolution 15x15 for each of its 128 filters

from tensorflow.keras.models import load_model

model = load_model(os.path.join(model_dir, 'cats_and_dogs_small_2.h5'))
model.summary()  # As a reminder.

Model: "sequential_2"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
conv2d_8 (Conv2D)            (None, 148, 148, 32)      896       
_________________________________________________________________
max_pooling2d_8 (MaxPooling2 (None, 74, 74, 32)        0         
_________________________________________________________________
conv2d_9 (Conv2D)            (None, 72, 72, 64)        18496     
_________________________________________________________________
max_pooling2d_9 (MaxPooling2 (None, 36, 36, 64)        0         
_________________________________________________________________
conv2d_10 (Conv2D)           (None, 34, 34, 128)       73856     
_________________________________________________________________
max_pooling2d_10 (MaxPooling (None, 17, 17, 128)       0         
_________________________________________________________________
conv2d_11 (Conv2D)           (None, 15, 15, 128)       147584    
_________________________________________________________________
max_pooling2d_11 (MaxPooling (None, 7, 7, 128)         0         
_________________________________________________________________
flatten_2 (Flatten)          (None, 6272)              0         
_________________________________________________________________
dropout_2 (Dropout)          (None, 6272)              0         
_________________________________________________________________
dense_4 (Dense)              (None, 512)               3211776   
_________________________________________________________________
dense_5 (Dense)              (None, 1)                 513       
=================================================================
Total params: 3,453,121
Trainable params: 3,453,121
Non-trainable params: 0
_________________________________________________________________

• To extract the activations, we create a new model that outputs the trained layers
  • 8 output layers in total (only the convolutional part)
• We input a test image for prediction and then read the relevant outputs

layer_outputs = [layer.output for layer in model.layers[:8]]
activation_model = models.Model(inputs=model.input, outputs=layer_outputs)
activations = activation_model.predict(img_tensor)

Output of the first Conv2D layer, 3rd channel (filter):

• Similar to a diagonal edge detector
• Your own channels may look different

plt.rcParams['figure.dpi'] = 120
first_layer_activation = activations[0]

f, (ax1, ax2) = plt.subplots(1, 2, sharey=True)
ax1.imshow(img_tensor[0])
ax2.matshow(first_layer_activation[0, :, :, 2], cmap='viridis')
ax1.set_xticks([])
ax1.set_yticks([])
ax2.set_xticks([])
ax2.set_yticks([])
ax1.set_xlabel('Input image')
ax2.set_xlabel('Activation of filter 2');

Output of filter 16:

• Cat eye detector?

f, (ax1, ax2) = plt.subplots(1, 2, sharey=True)
ax1.imshow(img_tensor[0])
ax2.matshow(first_layer_activation[0, :, :,16], cmap='viridis')
ax1.set_xticks([])
ax1.set_yticks([])
ax2.set_xticks([])
ax2.set_yticks([])
ax1.set_xlabel('Input image')
ax2.set_xlabel('Activation of filter 16');

The same filter responds quite differently for other inputs

img_path = os.path.join(base_dir, 'test/dogs/dog.1528.jpg')

# We preprocess the image into a 4D tensor
img = image.load_img(img_path, target_size=(150, 150))
img_tensor2 = image.img_to_array(img)
img_tensor2 = np.expand_dims(img_tensor2, axis=0) 
# Remember that the model was trained on inputs
# that were preprocessed in the following way:
img_tensor2 /= 255.

activations2 = activation_model.predict(img_tensor2)

plt.rcParams['figure.dpi'] = 120
first_layer_activation2 = activations2[0]

f, (ax1, ax2) = plt.subplots(1, 2, sharey=True)
ax1.imshow(img_tensor2[0])
ax2.matshow(first_layer_activation2[0, :, :, 16], cmap='viridis')
ax1.set_xticks([])
ax1.set_yticks([])
ax2.set_xticks([])
ax2.set_yticks([])
ax1.set_xlabel('Input image')
ax2.set_xlabel('Activation of filter 16');

• First 2 convolutional layers: various edge detectors

plot_activations(0, activations)
plot_activations(2, activations)

• 3rd convolutional layer: increasingly abstract: ears, eyes

plot_activations(4, activations)

• Last convolutional layer: more abstract patterns
• Empty filter activations: input image does not have the information that the filter was interested in

plot_activations(6, activations)

• Same layer, with dog image input
  • Very different activations

plot_activations(6, activations2)

Spatial hierarchies

• Deep convnets can learn spatial hierarchies of patterns
  • First layer can learn very local patterns (e.g. edges)
  • Second layer can learn specific combinations of patterns
  • Every layer can learn increasingly complex abstractions

Visualizing the learned filters

• The filters themselves can be visualized by finding the input image that they are maximally responsive to
• gradient ascent in input space: start from a random image, use loss to update the pixel values to values that the filter responds to more strongly

from keras import backend as K
    input_img = np.random.random((1, size, size, 3)) * 20 + 128.
    loss = K.mean(layer_output[:, :, :, filter_index])
    grads = K.gradients(loss, model.input)[0] # Compute gradient
    for i in range(40): # Run gradient ascent for 40 steps
        loss_v, grads_v = K.function([input_img], [loss, grads])
        input_img_data += grads_v * step

tf.compat.v1.disable_eager_execution()

# Convert tensor to image
def deprocess_image(x):
    # normalize tensor: center on 0., ensure std is 0.1
    x -= x.mean()
    x /= (x.std() + 1e-5)
    x *= 0.1

    # clip to [0, 1]
    x += 0.5
    x = np.clip(x, 0, 1)

    # convert to RGB array
    x *= 255
    x = np.clip(x, 0, 255).astype('uint8')
    return x

def generate_pattern(layer_name, filter_index, size=150):
    # Build a loss function that maximizes the activation
    # of the nth filter of the layer considered.
    layer_output = model.get_layer(layer_name).output
    loss = K.mean(layer_output[:, :, :, filter_index])

    # Compute the gradient of the input picture wrt this loss
    grads = K.gradients(loss, model.input)[0]

    # Normalization trick: we normalize the gradient
    grads /= (K.sqrt(K.mean(K.square(grads))) + 1e-5)

    # This function returns the loss and grads given the input picture
    iterate = K.function([model.input], [loss, grads])
    
    # We start from a gray image with some noise
    input_img_data = np.random.random((1, size, size, 3)) * 20 + 128.

    # Run gradient ascent for 40 steps
    step = 1.
    for i in range(40):
        loss_value, grads_value = iterate([input_img_data])
        input_img_data += grads_value * step
        
    img = input_img_data[0]
    return deprocess_image(img)

def visualize_filter(layer_name):
    size = 64
    margin = 5

    # This a empty (black) image where we will store our results.
    results = np.zeros((8 * size + 7 * margin, 8 * size + 7 * margin, 3))

    for i in range(8):  # iterate over the rows of our results grid
        for j in range(8):  # iterate over the columns of our results grid
            # Generate the pattern for filter `i + (j * 8)` in `layer_name`
            filter_img = generate_pattern(layer_name, i + (j * 8), size=size)

            # Put the result in the square `(i, j)` of the results grid
            horizontal_start = i * size + i * margin
            horizontal_end = horizontal_start + size
            vertical_start = j * size + j * margin
            vertical_end = vertical_start + size
            results[horizontal_start: horizontal_end, vertical_start: vertical_end, :] = filter_img

    # Display the results grid
    plt.figure(figsize=(10, 10))
    plt.imshow((results * 255).astype(np.uint8))

    plt.show()

• Learned filters of second convolutional layer
• Mostly general, some respond to specific shapes/colors

from tensorflow.keras import backend as K

plt.rcParams['figure.dpi'] = 60
visualize_filter('conv2d_9')

• Learned filters of last convolutional layer
• More focused on center, some vague cat/dog head shapes

visualize_filter('conv2d_11')

Let's do this again for the VGG16 network pretrained on ImageNet (much larger)

model = VGG16(weights='imagenet', include_top=False)

# VGG16 model
model.summary()

Model: "vgg16"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
input_1 (InputLayer)         [(None, None, None, 3)]   0         
_________________________________________________________________
block1_conv1 (Conv2D)        (None, None, None, 64)    1792      
_________________________________________________________________
block1_conv2 (Conv2D)        (None, None, None, 64)    36928     
_________________________________________________________________
block1_pool (MaxPooling2D)   (None, None, None, 64)    0         
_________________________________________________________________
block2_conv1 (Conv2D)        (None, None, None, 128)   73856     
_________________________________________________________________
block2_conv2 (Conv2D)        (None, None, None, 128)   147584    
_________________________________________________________________
block2_pool (MaxPooling2D)   (None, None, None, 128)   0         
_________________________________________________________________
block3_conv1 (Conv2D)        (None, None, None, 256)   295168    
_________________________________________________________________
block3_conv2 (Conv2D)        (None, None, None, 256)   590080    
_________________________________________________________________
block3_conv3 (Conv2D)        (None, None, None, 256)   590080    
_________________________________________________________________
block3_pool (MaxPooling2D)   (None, None, None, 256)   0         
_________________________________________________________________
block4_conv1 (Conv2D)        (None, None, None, 512)   1180160   
_________________________________________________________________
block4_conv2 (Conv2D)        (None, None, None, 512)   2359808   
_________________________________________________________________
block4_conv3 (Conv2D)        (None, None, None, 512)   2359808   
_________________________________________________________________
block4_pool (MaxPooling2D)   (None, None, None, 512)   0         
_________________________________________________________________
block5_conv1 (Conv2D)        (None, None, None, 512)   2359808   
_________________________________________________________________
block5_conv2 (Conv2D)        (None, None, None, 512)   2359808   
_________________________________________________________________
block5_conv3 (Conv2D)        (None, None, None, 512)   2359808   
_________________________________________________________________
block5_pool (MaxPooling2D)   (None, None, None, 512)   0         
=================================================================
Total params: 14,714,688
Trainable params: 14,714,688
Non-trainable params: 0
_________________________________________________________________

• Visualize convolution filters 0-2 from layer 5 of the VGG network trained on ImageNet
• Some respond to dots or waves in the image

for i in range(3):
    plt.subplot(131+i) 
    plt.xticks([]) 
    plt.yticks([])
    plt.imshow(generate_pattern('block3_conv1', i))
plt.tight_layout()
plt.show();

First 64 filters for 1st convolutional layer in block 1: simple edges and colors

plt.rcParams['figure.dpi'] = 60
visualize_filter('block1_conv1')

Filters in 2nd block of convolution layers: simple textures (combined edges and colors)

visualize_filter('block2_conv1')

Filters in 3rd block of convolution layers: more natural textures

visualize_filter('block3_conv1')

Filters in 4th block of convolution layers: feathers, eyes, leaves,...

visualize_filter('block4_conv1')

Visualizing class activation

• We can also visualize which part of the input image had the greatest influence on the final classification
  • Helpful for interpreting what the model is paying attention to
• Class activation maps: produce heatmap over the input image
  • Take the output feature map of a convolution layer (e.g. the last one)
  • Weigh every filter by the gradient of the class with respect to the filter

Illustration (cats vs dogs)

• These were the output feature maps of the last convolutional layer
  • These are flattened and fed to the dense layer
• Compute gradient of the 'cat' node output wrt. every filter output (pixel) here
  • Average the gradients per filter, use that as the filter weight
• Take the weighted sum of all filter maps to get the class activation map

plot_activations(6, activations2)

More realistic example:

• Try VGG (including the dense layers) and an image from ImageNet

  model = VGG16(weights='imagenet')
  

  

Preprocessing

• Load image
• Resize to 224 x 224 (what VGG was trained on)
• Do the same preprocessing (Keras VGG utility)

from keras.applications.vgg16 import preprocess_input
img_path = '../images/10_elephants.jpg'
img = image.load_img(img_path, target_size=(224, 224))
x = image.img_to_array(img)
x = np.expand_dims(x, axis=0) # Transform to batch of size (1, 224, 224, 3)
x = preprocess_input(x)

• Sanity test: do we get the right prediction?

preds = model.predict(x)

preds = model.predict(x)
print('Predicted:', decode_predictions(preds, top=3)[0])

Predicted: [('n02504458', 'African_elephant', 0.909421), ('n01871265', 'tusker', 0.086182885), ('n02504013', 'Indian_elephant', 0.0043545826)]

Visualize the class activation map

heatmap = np.maximum(heatmap, 0)
heatmap /= np.max(heatmap)
plt.matshow(heatmap)
plt.show()

Superimposed on the original image

img = cv2.imread('../images/elephant_cam.jpg')
RGB_im = cv2.cvtColor(img, cv2.COLOR_BGR2RGB)

plt.rcParams['figure.dpi'] = 120
plt.imshow(RGB_im)
plt.title('Class activation map')
plt.xticks([])
plt.yticks([])
plt.show()

Using pretrained networks

• We can re-use pretrained networks instead of training from scratch
• Learned features can be a generic model of the visual world
• Use convolutional base to contruct features, then train any classifier on new data
• Also called transfer learning, which is a kind of meta-learning

• Let's instantiate the VGG16 model (without the dense layers)
• Final feature map has shape (4, 4, 512)

  conv_base = VGG16(weights='imagenet', include_top=False, input_shape=(150, 150, 3))
  

conv_base.summary()

Model: "vgg16"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
input_2 (InputLayer)         [(None, 150, 150, 3)]     0         
_________________________________________________________________
block1_conv1 (Conv2D)        (None, 150, 150, 64)      1792      
_________________________________________________________________
block1_conv2 (Conv2D)        (None, 150, 150, 64)      36928     
_________________________________________________________________
block1_pool (MaxPooling2D)   (None, 75, 75, 64)        0         
_________________________________________________________________
block2_conv1 (Conv2D)        (None, 75, 75, 128)       73856     
_________________________________________________________________
block2_conv2 (Conv2D)        (None, 75, 75, 128)       147584    
_________________________________________________________________
block2_pool (MaxPooling2D)   (None, 37, 37, 128)       0         
_________________________________________________________________
block3_conv1 (Conv2D)        (None, 37, 37, 256)       295168    
_________________________________________________________________
block3_conv2 (Conv2D)        (None, 37, 37, 256)       590080    
_________________________________________________________________
block3_conv3 (Conv2D)        (None, 37, 37, 256)       590080    
_________________________________________________________________
block3_pool (MaxPooling2D)   (None, 18, 18, 256)       0         
_________________________________________________________________
block4_conv1 (Conv2D)        (None, 18, 18, 512)       1180160   
_________________________________________________________________
block4_conv2 (Conv2D)        (None, 18, 18, 512)       2359808   
_________________________________________________________________
block4_conv3 (Conv2D)        (None, 18, 18, 512)       2359808   
_________________________________________________________________
block4_pool (MaxPooling2D)   (None, 9, 9, 512)         0         
_________________________________________________________________
block5_conv1 (Conv2D)        (None, 9, 9, 512)         2359808   
_________________________________________________________________
block5_conv2 (Conv2D)        (None, 9, 9, 512)         2359808   
_________________________________________________________________
block5_conv3 (Conv2D)        (None, 9, 9, 512)         2359808   
_________________________________________________________________
block5_pool (MaxPooling2D)   (None, 4, 4, 512)         0         
=================================================================
Total params: 14,714,688
Trainable params: 14,714,688
Non-trainable params: 0
_________________________________________________________________

Using pre-trained networks: 3 ways

• Fast feature extraction (similar task, little data)
  • Call predict from the convolutional base to build new features
  • Use outputs as input to a new neural net (or other algorithm)
• End-to-end tuning (similar task, lots of data + data augmentation)
  • Extend the convolutional base model with a new dense layer
  • Train it end to end on the new data (expensive!)
• Fine-tuning (somewhat different task)
  • Unfreeze a few of the top convolutional layers, and retrain
    • Update only the more abstract representations

Fast feature extraction (without data augmentation)

• Run every batch through the pre-trained convolutional base

generator = datagen.flow_from_directory(dir, target_size=(150, 150),
        batch_size=batch_size, class_mode='binary')
for inputs_batch, labels_batch in generator:
    features_batch = conv_base.predict(inputs_batch)

• Build Dense neural net (with Dropout)
• Train and evaluate with the transformed examples

model = models.Sequential()
model.add(layers.Dense(256, activation='relu', input_dim=4 * 4 * 512))
model.add(layers.Dropout(0.5))
model.add(layers.Dense(1, activation='sigmoid'))

• Validation accuracy around 90%, much better!
• Still overfitting, despite the Dropout: not enough training data

history = pickle.load(open("../data/models/cats_and_dogs_small_3a_history.p", "rb"))
print("Max val_acc",np.max(history['val_acc']))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Max val_acc 0.9070000052452087

Fast feature extraction (with data augmentation)

• Simply add the Dense layers to the convolutional base
• Freeze the convolutional base (before you compile)
  • Without freezing, you train it end-to-end (expensive)

model = models.Sequential()
model.add(conv_base)
model.add(layers.Flatten())
model.add(layers.Dense(256, activation='relu'))
model.add(layers.Dense(1, activation='sigmoid'))
conv_base.trainable = False

model.summary()

Model: "sequential_3"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
vgg16 (Model)                (None, 4, 4, 512)         14714688  
_________________________________________________________________
flatten_3 (Flatten)          (None, 8192)              0         
_________________________________________________________________
dense_6 (Dense)              (None, 256)               2097408   
_________________________________________________________________
dense_7 (Dense)              (None, 1)                 257       
=================================================================
Total params: 16,812,353
Trainable params: 2,097,665
Non-trainable params: 14,714,688
_________________________________________________________________

Data augmentation and training (takes a LONG time)

train_datagen = ImageDataGenerator(
      rescale=1./255, rotation_range=40, width_shift_range=0.2,
      height_shift_range=0.2, shear_range=0.2, zoom_range=0.2,
      horizontal_flip=True, fill_mode='nearest')
train_generator = train_datagen.flow_from_directory(dir,
      target_size=(150, 150), batch_size=20, class_mode='binary')
history = model.fit_generator(
      train_generator, steps_per_epoch=100, epochs=30,
      validation_data=validation_generator, validation_steps=50)

We now get about 90% accuracy again, and very little overfitting

history = pickle.load(open("../data/models/cats_and_dogs_small_3b_history.p", "rb"))
print("Max val_acc",np.max(history['val_acc']))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Max val_acc 0.906

Fine-tuning

• Add your custom network on top of an already trained base network.
• Freeze the base network, but unfreeze the last block of conv layers.

for layer in conv_base.layers:
    if layer.name == 'block5_conv1':
        layer.trainable = True
    else:
        layer.trainable = False

Visualized

• Load trained network, finetune
  • Use a small learning rate, large number of epochs
  • You don't want to unlearn too much: catastrophic forgetting

model = load_model(os.path.join(model_dir, 'cats_and_dogs_small_3b.h5'))
model.compile(loss='binary_crossentropy', 
              optimizer=optimizers.RMSprop(lr=1e-5),
              metrics=['acc'])
history = model.fit(
      train_generator, steps_per_epoch=100, epochs=100,
      validation_data=validation_generator,
      validation_steps=50)

Almost 95% accuracy. The curves are quite noisy, though.

history = pickle.load(open("../data/models/cats_and_dogs_small_3c_history.p", "rb"))
print("Max val_acc",np.max(history['val_acc']))
pd.DataFrame(history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Max val_acc 0.948

• We can smooth the learning curves using a running average

def smooth_curve(points, factor=0.8):
  smoothed_points = []
  for point in points:
    if smoothed_points:
      previous = smoothed_points[-1]
      smoothed_points.append(previous * factor + point * (1 - factor))
    else:
      smoothed_points.append(point)
  return smoothed_points

smooth_history = {}
smooth_history['loss'] = smooth_curve(history['loss'])
smooth_history['acc'] = smooth_curve(history['acc'])
smooth_history['val_loss'] = smooth_curve(history['val_loss'])
smooth_history['val_acc'] = smooth_curve(history['val_acc'])

print("Max val_acc",np.max(smooth_history['val_acc']))
pd.DataFrame(smooth_history).plot(lw=2,style=['b:','r:','b-','r-']);
plt.xlabel('epochs');

Max val_acc 0.9408402440315202

Finally, evaluate the trained model on the test set. This is consistent with the validation results.

test_generator = test_datagen.flow_from_directory(
        test_dir,
        target_size=(150, 150),
        batch_size=20,
        class_mode='binary')

model = load_model(os.path.join(model_dir, 'cats_and_dogs_small_3c.h5'))
test_loss, test_acc = model.evaluate(test_generator, steps=50)
print('test acc:', test_acc)

Found 1000 images belonging to 2 classes.
50/50 [==============================] - 126s 3s/step - loss: 0.3771 - acc: 0.9390
test acc: 0.939

Take-aways

• Convnets are ideal for attacking visual-classification problems.
• They learn a hierarchy of modular patterns and concepts to represent the visual world.
• Representations are easy to inspect
• Data augmentation helps fight overfitting
• You can use a pretrained convnet to build better models via transfer learning